Modeling a Single Conic Section

• Sound propagation in conical air columns can be well modeled by one-dimensional spherical waves traveling along the length of the cone.

• The one-dimensional wave equation for spherical pressure waves,
    $$\frac{1}{x} \ensuremath{\frac{\partial^{2} {(xp)}}{\partial {x}^{... ... = \frac{1}{c^{2}} \ensuremath{\frac{\partial^{2} {p}}{\partial {t}^{2}}} \, ,$$ (20)
  accurately represents lossless pressure wave propagation along the central axis of a conical tube, subject to the boundary conditions at both its ends.

• The continuous-time traveling-wave solution to this equation is
    $$p(x,t) = \frac{f(t - x/c)}{x} + \frac{g(t + x/c)}{x} \, ,$$ (21)
  where the functions $f(\cdot)$ and $g(\cdot)$ are completely general and continuous and can be interpreted as arbitrarily fixed waveshapes that travel in opposite directions along the $x$-axis with speed $c.$

• This expression is similar to that for plane waves with the exception that spherical pressure traveling-wave components are inversely proportional to their distance from the cone apex.

• A solution of this form can be discretized in time and space and given by (Smith, 1991)
    $$p\left(t_{n},x_{m}\right) = \frac{p^{+}(n - m)}{x} + \frac{p^{-}(n + m)}{x}.$$ (22)

• The behavior of a finite length conical bore can be approximated for low-frequency sound waves by assuming that pressure is equal to zero at an open end. This boundary condition is met with an inversion of traveling-wave pressure components at the open end.

• Figure 4 represents the digital waveguide implementation of ideal, lossless spherical-wave propagation in an ideally terminated conical tube.

  Figure 4: Digital waveguide implementation of ideal, lossless spherical-wave propagation in a conical tube.

  

• Aside from the $1/x$ scale factors, which are implemented at observation points, the cylindrical and conical waveguide implementations are exactly the same.

• Further, if the system input and output are measured at the same location, the $1/x$ scale factor is unnecessary.